Results: Plots and Tables

This report contains different plots and tables that may be relevant for analysing the results. Observe:

• The hypercube \(Y_N\) is placed in, is approx. \([0, 10000m]^p\), since for each subproblem \(Y_N^s \subset [0, 10000]^p\). That is, more subproblems create a larger hypercube for the master problem and the volume (number of integer points) grows exponentially in \(p\).
• Given \(|Y_N|\), we can find the hypercube as \[[l_1, u_1] \times \cdots \times [l_p, u_p],\] where \(l_i = \min_{y \in Y_N}(y_i)\) and \(u_i = \max_{y \in Y_N}(y_i)\).
• The number of integer points in the hypercube is \[\prod_{i = 1}^p (u_i - l_i + 1)\]

Statistics for alg1

Given a problem consisting of \(m\) subproblems with \(Y_N^s\) given for each subproblem \(s\), we use a filtering algorithm to find \(Y_N\) (alg1).

The following instance/problem groups are generated given:

• \(p=2,\ldots, 5\). [4 options]
• \(m=2,\ldots 5\) where \(m\) is the number of subproblems. [4 options]
• All subproblems have the same method config or half have method u and l. [4 options]
• Five instances for each config. [5 options]

Status

• 822/1600 problems have been solved, i.e. 778 remaining.
• 820/822 problems have 5 instances solved for each configuration.
• 255/822 have not been classified.
• 48/822 have not been fully classified (only classified extreme).

Problems solved for the analysis

Note that the width of objective \(i = 1, \ldots p\), \(w_i = [l_i, u_i]\) should be approx. \(10000m\). Check:

## # A tibble: 4 × 6
##       m mean_width1 mean_width2 mean_width3 mean_width4 mean_width5
##   <dbl>       <dbl>       <dbl>       <dbl>       <dbl>       <dbl>
## 1     2      19245.      19221.      19213.      18996.      18690.
## 2     3      28635.      28677.      28545.      28414.      27663.
## 3     4      38098.      38229.      38367.      37928.      37041.
## 4     5      47125.      47586.      47515.      46924.      44537.

Size of \(Y_N\)

What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?

## # A tibble: 4 × 3
##   method mean_card     n
##   <chr>      <dbl> <int>
## 1 l         11575.   115
## 2 m        411493.   295
## 3 u        104870.   295
## 4 ul         7352.   115

Does \(p\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     p mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     1848.    30
##  2 m          2     5528.    75
##  3 u          2     1059.    75
##  4 ul         2      623.    30
##  5 l          3     6210.    30
##  6 m          3   105312.    75
##  7 u          3    10077     75
##  8 ul         3     3016.    30
##  9 l          4    18485.    30
## 10 m          4   476291.    75
## 11 u          4    79341.    75
## 12 ul         4    11504.    30
## 13 l          5    21395.    25
## 14 m          5  1105081.    70
## 15 u          5   345012.    70
## 16 ul         5    15647.    25

Does \(m\) have an effect?

## # A tibble: 12 × 4
## # Groups:   method [4]
##    method     m mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     8173.    80
##  2 m          2     5688.    80
##  3 u          2     4201.    80
##  4 ul         2     4923.    80
##  5 l          3    19352.    35
##  6 m          3    90077.    80
##  7 u          3    37283.    80
##  8 ul         3    12904.    35
##  9 m          4   874692.    80
## 10 u          4   190675.    80
## 11 m          5   795527.    55
## 12 u          5   224798.    55

Relative size of \(Y_N\)

Nondominated points classification

We classify the nondominated points into, extreme, supported non-extreme and unsupported.

Plots used in the paper